The 19-year-old, who has recently returned from a loan spell with Tranmere Rovers, is expected to head out on loan again before the end of the transfer window.
The Swedish winger told Fotballskanalen.se: "It feels terrific. It is actually kind of a shock. A few months ago I didn't believe this would happen.
"Brendan [Rodgers] wanted me to sign a contract as soon as possible. It was me and two others who got new contracts. This is the biggest motivation boost in my career. This is the biggest I have ever experienced.
"Of course, I am aiming for the first team. Liverpool would not have offered me a contract in case they did not have confidence in me. The first step is to go on-loan as soon as possible, I want to play first team football. Then if Liverpool have other plans for me, it is obvious that I will stay."
Peterson, who joined the club as a 16-year-old, is yet to make his first-team debut for the Anfield side.